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Theorem isfin5 8179
Description: Definition of a V-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
isfin5  |-  ( A  e. FinV  <-> 
( A  =  (/)  \/  A  ~<  ( A  +c  A ) ) )

Proof of Theorem isfin5
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-fin5 8169 . . 3  |- FinV  =  {
x  |  ( x  =  (/)  \/  x  ~<  ( x  +c  x
) ) }
21eleq2i 2500 . 2  |-  ( A  e. FinV  <-> 
A  e.  { x  |  ( x  =  (/)  \/  x  ~<  (
x  +c  x ) ) } )
3 id 20 . . . . 5  |-  ( A  =  (/)  ->  A  =  (/) )
4 0ex 4339 . . . . 5  |-  (/)  e.  _V
53, 4syl6eqel 2524 . . . 4  |-  ( A  =  (/)  ->  A  e. 
_V )
6 relsdom 7116 . . . . 5  |-  Rel  ~<
76brrelexi 4918 . . . 4  |-  ( A 
~<  ( A  +c  A
)  ->  A  e.  _V )
85, 7jaoi 369 . . 3  |-  ( ( A  =  (/)  \/  A  ~<  ( A  +c  A
) )  ->  A  e.  _V )
9 eqeq1 2442 . . . 4  |-  ( x  =  A  ->  (
x  =  (/)  <->  A  =  (/) ) )
10 id 20 . . . . 5  |-  ( x  =  A  ->  x  =  A )
1110, 10oveq12d 6099 . . . . 5  |-  ( x  =  A  ->  (
x  +c  x )  =  ( A  +c  A ) )
1210, 11breq12d 4225 . . . 4  |-  ( x  =  A  ->  (
x  ~<  ( x  +c  x )  <->  A  ~<  ( A  +c  A ) ) )
139, 12orbi12d 691 . . 3  |-  ( x  =  A  ->  (
( x  =  (/)  \/  x  ~<  ( x  +c  x ) )  <->  ( A  =  (/)  \/  A  ~<  ( A  +c  A ) ) ) )
148, 13elab3 3089 . 2  |-  ( A  e.  { x  |  ( x  =  (/)  \/  x  ~<  ( x  +c  x ) ) }  <-> 
( A  =  (/)  \/  A  ~<  ( A  +c  A ) ) )
152, 14bitri 241 1  |-  ( A  e. FinV  <-> 
( A  =  (/)  \/  A  ~<  ( A  +c  A ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    \/ wo 358    = wceq 1652    e. wcel 1725   {cab 2422   _Vcvv 2956   (/)c0 3628   class class class wbr 4212  (class class class)co 6081    ~< csdm 7108    +c ccda 8047  FinVcfin5 8162
This theorem is referenced by:  isfin5-2  8271  fin56  8273
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-iota 5418  df-fv 5462  df-ov 6084  df-dom 7111  df-sdom 7112  df-fin5 8169
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